Intersection of Parametric Surfaces by Means of Look-Up Tables
IEEE Computer Graphics and Applications
A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
Computer Aided Geometric Design
Direct boolean intersection between acquired and designed geometry
Computer-Aided Design
On NURBS algorithms using tangent cones
Computer Aided Geometric Design
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Intersection of surfaces is a recurring problem in CAD/CAM and geometric modelling. Such intersections may yield any combination of curves and isolated points. Computation using analytic forms requires n(n - 1)/2 algorithms for n surface equations. Use of parametric forms of the surfaces allows development of a single algorithm dependent only on availability of parametric surface evaluators. The algorithm comprises four steps. First, the surfaces are subdivided using an iterative subdivision process. The subdivision criteria are the curvature and the boundary linearity of local subpieces. Oriented rectangular parallelepipeds cull subpiece pairs which are clearly disjoint. The second step is intersection of subpiece pairs. The subpieces (flat within the subdivision limits) are each approximated by two triangles. The intersection of these triangles in pairs yields a maximum of four linear segments. The third step is a sorting process, using subdivision trees to connect the segments. The final step is refinement of the intersection points. A Newton-Raphson like method improves the placement of the segment endpoints on the two surfaces. The selectivity of subdivision, and placement of refinement after location and sorting serve to reduce evaluations and hence running time for the algorithm. Design and implementation considerations and problems are discussed and some run-times are presented.